2 Answers
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The Webmasters FAQ outlines the nature of acceptable questions for this site: researching available solutions and providing anecdotal advice falls outside the purpose of this StackExchange site, so this type of question would be better-suited to chat or another discussion forum.

That being said, there are a variety of free and open source content management systems, as well as licensed software and paid services, which may meet your requirements.

All of the forum software listed above are built around the concept of mods (plugins) which extend the application's functionality - there are many mods, with new mods authored every day.

Sites like forum-software.org can help you compare different forum software, and have forum comparison tools and in-depth reviews.

If you have researched different forum software and available mods but cannot find anything that will meet your needs, you should consult a developer (or do it yourself) who can build something to your specifications either as a mod for an existing forum script or as a stand-alone application.

I like the Xenforo forum software. It's built by these guys who left VBulliten and decided to try and build a cooler forum software. It's sleeker and looks newer. It doesn't have as many extensions seeing as it's only a couple years old. But, Xenforo definitely worth checking out.

Consider merging your answer with the one above, as your points will get more attention that way. Also, I removed the link to your site: it isn't relevant to this answer.
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ChristofianMar 5 '12 at 1:36